In this paper, by using the Leggett-Williams norm-type theorem, we consider a m-point boundary value problem for a class of fractional differential equations at resonance. A new result on the existence of solutions for above problem is obtained.

The subject of fractional calculus has gained significant interest and been a valuable tool for both science and engineering (see [1–3]). In recent years, the fractional boundary value problems (FBVPs for short) have been considered by many authors (see [4–10] and the references therein). For example, Bai studied a FBVP at non-resonance with \(1<\alpha\leq2\) (see [10]). FBVPs at resonance were studied by Kosmatov (see [11]) and Jiang (see [12]). But the positive solutions for FBVPs at resonance were studied very few. In [13], Yang and Wang considered the positive solutions of the following FBVP:

However, to the best of our knowledge, the fractional differential equations with m-point boundary conditions at resonance have not been considered. Motivated by the papers above, we consider the existence of positive solutions for a m-point FBVP of the form

The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions and lemmas. In Section 3, we establish a theorem on the existence of positive solutions for FBVP (1.1) under some restrictions of f, basing on the coincidence degree theory due to [15]. Finally, in Section 4, an example is given to illustrate the main result.

is invertible. We denote the inverse by \(K_{P}\). Moreover, by virtue of \(\operatorname{dim} \operatorname{Im}Q=\operatorname{codim} \operatorname{Im}L\), there exists an isomorphism \(J: \operatorname{Im}Q\rightarrow\operatorname{Ker}L\). Then we know that the operator equation \(Lx=Nx\) is equivalent to

$$x=(P+JQN)x+K_{P}(I-Q)Nx, $$

where \(N: X\rightarrow Y\) be a nonlinear operator.

If Ω is an open bounded subset of X such that \(\operatorname{dom} L\cap\overline{\Omega} \neq\emptyset\), then the map \(N:X\rightarrow Y\) will be called L-compact on \(\overline{\Omega}\) if \(QN:\overline {\Omega}\rightarrow Y\) is bounded and \(K_{P}(I-Q)N:\overline{\Omega }\rightarrow X\) is compact.

Lemma 2.2

LetCbe a cone inXand\(\Omega_{1}\), \(\Omega_{2}\)be open bounded subsets ofXwith\(\overline{\Omega}_{1} \subset\Omega_{2}\)and\(C\cap (\overline{\Omega}_{2}\setminus\Omega_{1})\neq\emptyset\). Assume that the following conditions are satisfied:

Proof

Obviously, \(\operatorname{Im} P=\operatorname{Ker} L\) and \(P^{2}x=Px\). It follows from \(x=(x-Px)+Px\) that \(X=\operatorname{Ker} P+\operatorname {Ker} L\). By a simple calculation, one obtain \(\operatorname{Ker} L\cap\operatorname{Ker} P=\{0\}\). Thus, we get

Let \(y=(y-Qy)+Qy\), where \(y-Qy\in\operatorname{Ker} Q\), \(Qy\in\operatorname{Im} Q\). It follows from \(\operatorname{Ker} Q=\operatorname{Im} L\) and \(Q^{2}y=Qy\) that \(\operatorname{Im} Q\cap\operatorname{Im} L=\{ 0 \}\). Then one has

Combining (3.6) with (3.7), we know that \(K_{P}=(L|_{\operatorname {dom}L\cap\operatorname{Ker}P})^{-1}\). The proof is complete. □

Lemma 3.4

Assume\(\Omega\subset X\)is an open bounded subset such that\(\operatorname{dom}L\cap\overline{\Omega}\neq\emptyset\), thenNisL-compact on\(\overline{\Omega}\).

Proof

By the continuity of f, we see that \(QN(\overline{\Omega})\) and \(K_{P}(I-Q)N(\overline{\Omega})\) are bounded. That is, there exist constants \(A,B>0\) such that \(|(I-Q)Nx|\leq A\) and \(|K_{P}(I-Q)Nx|\leq B\), \(\forall x\in\overline{\Omega}\), \(t\in[0,1]\). Thus, one need only prove that \(K_{P}(I-Q)N(\overline{\Omega})\subset X\) is equicontinuous.

Let \(K_{P,Q}=K_{P}(I-Q)N\), for \(0\leq t_{1}< t_{2}\leq1\), \(x\in\overline {\Omega}\), we get

Since \(t^{\alpha}\) is uniformly continuous on \([0,1]\), we see that \(K_{P,Q}N(\overline{\Omega})\subset X\) is equicontinuous. Thus, we see that \(K_{P,Q}N:\overline{\Omega}\rightarrow X\) is compact. The proof is completed. □

From Lemma 3.3, Lemma 3.4, and Lemma 3.5, we see that the conditions (1) and (2) of Lemma 2.2 are satisfied.

Let \(\gamma x(t)=|x(t)|\) for \(x\in X\) and \(J=I\). One can see that γ is a retraction and maps subsets of \(\overline{\Omega}_{2}\) into bounded subsets of C, which means that the condition (3) of Lemma 2.2 holds.

For \(x\in\operatorname{Ker}L \cap\Omega_{2}\), we have \(x(t)\equiv c\). Let

Thus, the condition (7) of Lemma 2.2 holds. In addition, we can prove the condition (6) of Lemma 2.2 holds too by a similar process.

Finally, we will show that the condition (5) of Lemma 2.2 is satisfied. Let \(u_{0}(t)\equiv1\), \(t\in[0,1]\), then \(u_{0} \in C\setminus\{ 0\}\), \(C(u_{0})=\{ x\in C: x(t)>0, t\in[0,1]\}\) and we can take \(\sigma (u_{0})=1\). For \(x\in C(u_{0})\cap\partial\Omega_{1}\), we have \(x(t)>0\), \(t\in[0,1]\), \(0<\|x\|_{\infty}\leq r\), and \(x(t)\geq M\|x\|_{\infty}\), \(t\in[0,1]\). So, from (H4), we obtain

Consequently, by Lemma 2.2, the equation \(Lx=Nx\) has at least one solution \(x^{*}\in C\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\). Namely, FBVP (1.1) has at least one positive solution in X. The proof is complete. □

Moreover, \(f(t,u)\geq8-\frac{1}{10}u\geq-\frac{1}{4}u\) for all \(u\geq 0\), and \(l(s)\leq1\), \(G(t,s)\leq4\), \(\kappa=-\frac{1}{4}\). So, we can find that (H1), (H2), (H3) hold. Next, we take \(t_{0}=0\), \(h(x)=x\), and \(M=\frac{2}{3}\), thus \(G(0,s)= \frac{1}{\Gamma(\frac{5}{2})}(1-s)^{\frac{3}{2}}+\frac{3(\Gamma(\frac {7}{2})-1)}{2\Gamma(\frac{7}{2})(1-(\frac{1}{2})^{\frac{3}{2}})}l(s)\), \(0\leq s \leq1\), and \(\int_{0}^{1}G(0,s)\,ds=1\). Then (H4) is satisfied. According to the above points, by Theorem 3.1, we can conclude that FBVP (4.1) has at least one positive solution.

Acknowledgements

The authors would like to thank the referees very much for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (11271364).

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.